There is no evidence that singularities actually exist in nature. The original meaning of "singularity" in a math-science context is a breakdown in a man-made theory. A place where the mathematical model we are using fails and stops giving meaningful numbers for answers.
Often a singularity is where the man-made mathematical model "blows up" and gives no number at all (infinite density, infinite curvature...)
We do not infer from that that NATURE blows up
That's only the impression given in popularizations, magazine articles, bestseller pop books.
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There's probably something about that at Einstein-Online (link in my sig). But the main thing is that in the history of physics there have been lots of singularities (breakdowns blowups) and they have been FIXED by improving the theory and then checking the predictions of the improved theory.
So Einstein 1915 GR is the theory which, in this case, has a singularity in the Schwarzschild BH model (circa 1920). And the professional reaction is to admit that
we don't know what happens at the place where the classic unquantized theory breaks down
And we just have to wait until people fix the theory so it does not have a singularity, and then test the revised theory to be sure it makes more accurate predictions.
There are already (QG, quantized General Rel, quantum geometry) BH models.
There are BH models which do not have the singularity. Some of the people who have done the theoretical work are Kevin Vandersloot, Leonardo Modesto, Dah-wei Chiou.
But the work has not been observationally tested and AFAIK, the testing experts (phenomenology people) haven't taken the job over.
Work is much more advanced on getting rid of the cosmological singularity. The corresponding thing with BHs seems to be temporarily on the back burner. So we just have to wait and see.